A particle moves rectilinearly with initial velocity u and constant acceleration a. Find the average velocity of the particle in a time interval from `t=0` to `t=t` second of its motion.

Explanation

To find the average velocity of a particle moving rectilinearly with an initial velocity \( u \) and constant acceleration \( a \) over a time interval from \( t = 0 \) to \( t = t \) seconds, we can follow these steps:

### Step 1: Understand the formula for average velocity
The average velocity \( v_{avg} \) is defined as the total displacement \( s \) divided by the total time \( t \):
\[ v_{avg} = \frac{s}{t} \]

### Step 2: Determine the expression for total displacement
The total displacement \( s \) of the particle can be calculated using the kinematic equation:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( u \) is the initial velocity,
- \( a \) is the constant acceleration,
- \( t \) is the time interval.

### Step 3: Substitute the expression for displacement into the average velocity formula
Now, we substitute the expression for \( s \) into the average velocity formula:
\[ v_{avg} = \frac{ut + \frac{1}{2} a t^2}{t} \]

### Step 4: Simplify the expression
We can simplify the equation:
\[ v_{avg} = \frac{ut}{t} + \frac{\frac{1}{2} a t^2}{t} \]
This simplifies to:
\[ v_{avg} = u + \frac{1}{2} a t \]

### Final Result
Thus, the average velocity of the particle over the time interval from \( t = 0 \) to \( t = t \) seconds is:
\[ v_{avg} = u + \frac{1}{2} a t \]

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